Recombination is a major force in HIV evolution, but it is hard to study. It is particularly challenging to identify intra-host recombination between closey related viruses. The Investigators therefore propose new mathematical methods to study recombination in HIV and reticulate evolution generally. Each method will address biologically relevant questions about HIV infection and epidemiology. This proposal contributes to topological data analysis (TDA), a new quantitative approach to find structure in large datasets. The Investigators recently found that TDA provides rich information about evolving populations and is useful for detecting recombination. Objectives: Aim 1: Create new methods in TDA to characterize recombination; Aim 2: Apply these methods to HIV evolution; Aim 3: Build software tools for the research community. Aims will proceed iteratively, and preliminary results enable Aims 1 & 2 to begin immediately. The high-level strategy for each evolutionary scenario is: (1) Simulate specific scenario, (2) Discover topological statistic(s) corresponding to biological quantities of interest, (3) Validate statistic's power to estimate biological quantity in simulation, (4) Explain/Prove why the statisti is a valid estimator, (5) Apply statistic to biological/clinical questions.